Phase diagrams and shape transformations of toroidal vesicles

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Phase diagrams and shape transformations of toroidal vesicles

Frank Jülicher, Udo Seifert, Reinhard Lipowsky

To cite this version:

Frank Jülicher, Udo Seifert, Reinhard Lipowsky. Phase diagrams and shape transformations of toroidal vesicles. Journal de Physique II, EDP Sciences, 1993, 3 (11), pp.1681-1705. �10.1051/jp2:1993225�.

�jpa-00247932�

J. Phys. II FYance 3

(1993)

1681-1705 NOVEMBER1993, PAGE 1681

Classification Physics Abstracts 82.70

Phase diagrams and shape transformations of toroidal vesicles

Frank

Jiilicher,

Udo Seifert and Reinhard

Lipowsky

Institut fur Festk6rperforschung, Forschungszentrum Jiifich, 52425 Jiifich, Germany

(Received

10 May 1993, accepted in final form 21 July

1993)

Abstract. Shapes of vesicles with toroidal topology _are studied in the context of curvature models for the membrane. For two

simplified

curvature models, the spontaneous-curvature

(SC)

model and the bilayer-couple

(BC)

model, the structure of _energy diagrams, sheets of

stationary shapes and phase diagrams are obtained by solving shape equations for axisymmetric shapes. Three different sheets of axisymmetric shapes _are investigated systematically: I) discoid tori;

it)

sickle-shaped tori and iii) toroidal stomatocytes. ^A stability analysis of axisymmetric shapes with respect to symmetry breaking conformal transformations reveals that large regions of the phase diagrams of toroidal vesicles _are non-axisymmetric. Non-axisymmetric shapes

are determined approxim~tely using conformal transformations. To compare the theory with experiments, _a generalization of the SC and BC model, the area-difference-elasticity-model

(ADlGmodel),

which is

a more realistic curvature model for lipid bilayers, is discussed. Shapes of toroidal vesicles which have been observed recently _can be located in the phase diagram of

the ADlGmodel. We predict the effect oftemperature changes _on the observed shapes-The _new

class of shapes, the toroidal stomatocytes, have not yet been observed.

1. Introduction.

Vesicles _are closed

surfaces,

formed

by lipid bilayers

in aqueous solution

[1-3].

^The most

general

classification of such closed surfaces is

by

their

topology.

The different

topological

classes _are

distinguished by

their _genus, that is

by

the number of handles

(or holes).

Vesicles of

spherical topology

have been studied

intensively,

_e-g- in

experiments,

where _a

change

in temperature or

osmotic conditions induces

shape

transformations between _a

variety

of different

shapes [4-6].

Recently,

toroidal vesicles have been observed

experimentally.

The first

examples

_were vesi- cles made of _a

partially polymerized bilayer

_[7].

Typically,

these vesicles _were close _{to one}

special axisymmetric

circular

shape,

the sc-called Clifford _torus.

Subsequently,

toroidal vesi- cles

consisting

of fluid membranes have been found [8, 9].

They

exhibit _a

variety

of different types of

shapes

of genus one, two and _even

higher

_genus.

Among

the

genus-1

surfaces observed

so

far, axisymmetric

and non-axisymmetric

shapes

_occur. The genus-2 ^{surfaces which} are

necessarily non-axisymmetric

_can

crudly

be classified into two classes: I) ^discoid

shapes

with two holes

and11) shape3 consisting

of two

spherical

parts ^connected

by

three necks [8,

9].

Theoretical models for vesicles _are based _on the idea that the

shape

of _a vesicle is determined

by bending elasticity

which _can be

expressed by

the curvature of the surface. For

spherical vesicles,

the

phase diagrwas

of two curvature models

[lo, _11],

the spontaneous-curvature

(SC)

model and the

bilayer-couple (BC)

model have been studied

systematically [12-14].

These

phase diagrams

are obtained

by minimizing

the

bending

_energy for

axisymmetric shapes

with

a

given

surface _area and enclosed volume.

Recently,

_{a more} realistic

model,

the area~difference-

elasticity-model (ADE-model)

has been

proposed

which takes into account that _a

bilayer

is

composed

of two

compressible monolayers

^which can

glide

_over each other

[15-18].

^{The SC and}

the BC model turn out to be mathematical limits of this _more

general

model.

The

systematic

theoretical

study

of toroidal vesicles of

genus-I

started with _a minimal model for symmetric

bilayers (without

spontaneous

curvature)

with the volume to _area ratio, I-e- the reduced

volume,

_as the

only

parameter. ^Similar to the

spherical topology,

several branches of

axisymmetric shapes

were found and _a one-dimensional

phase diagram

was determined

[19, 20].

According

to _a famous

conjecture

the

axisymmetric shape

of lowest _energy

irrespective

of the reduced

volume,

the Clifford torus, ^is an absolute _energy minimum

[21].

^{Since the}

bending

energy is _a conformal

invariant,

this

ground

state is

degenerate,

I-e- _a one-parameter

family

of

non-axisymmetric shapes

with the _same energy can be

generated

from the Clifford torus

by applying

^conformal transformations [22]. ^The

physical

constraint of _a fixed volume _{to area} ratio breaks this

degeneracy. Consequently,

the

ground

state at fixed reduced volume is

unique

but

non-axisymmetric

over a

large

_range in the reduced volume

[20, 23].

^{In the}

vicinity

^{of the}

Clifford torus, ^the

eigenmodes

of the

bending

_energy which lead _to

axisymmetry breaking

have been calculated

analytically _{[24, 25].}

In this _{paper, we} aim for _a

thorough

theoretical

analysis

^of toroidal vesicle

shapes

from which _statements about recent

experimental

observations _as well _as

predictions

^{for future}

experiments

can be infered. We start in section 2

by analysing

the _{two curvature}

models,

the SC and the BC

model,

which both have been studied

intensively

for vesicles of

spherical topology.

We

classify

the sheets [26] ^of

stationary shapes

and determine the

phase diagrams

for both models. Even

though,

_as outlined in section 3, there _are

good

_reasons to believe that neither of the two models

gives

_a faithful

description

of the

bilayer

membrane, _a detailed

study

of these

(from

_our present

perspective) "simplified"

_or "mathematical" curvature

models,

is

justified.

First of

all, important insight

into the structure of the

possible shapes

can

already

be obtained in these models.

Secondly,

the

phase diagram

for _{a more} realistic

model,

the

area-difference-elasticity (ADE)

model _as discussed in section

3.1,

_can

easily

be derived

by

_a

general Legendre

transformation from the

phase diagram

^{of the BC} model. The

phase diagrwa

of the

ADE-model,

then

provides

^{the basis for} a

comparison

^with recent

experiments

in section 3.2.

Moreover,

_{we can} make

predictions

for future

experiments

^{in section} 3.3.

To facilitate _a coherent

reading

of the _{paper, we}

give

in section 2

only

results while

relegating

all

important

technicalities to the

appendices.

The

shape equations

for toroidal

topology

are

derived in

appendix

A. In

appendix B,

_we discuss in detail the _energy

diagrams

for

stationary shapes

in the SC and the BC model _as well _as the different types of limit

shapes.

In

appendix C,

the

stability

criterion of

axisymmetric shapes

with respect to symmetry

breaking

conformal

transformations is derived.

N°11 SHAPE TRANSFORMATIONS OF TOROmAL VESICLES 1683

2.

Simplified

curvature models.

2.I SPONTANEOUS-CURVATURE _MODEL. In the SC model of

Helfrich,

the _energy of the

bilayer

is

given by

[10]

F _e

f / _dA(Ci

+ C2

Co)~

+ _~cg

/ _{dACiC2 (1)}

2

Here, Ci

and C2 denote the

principal

curvatures on the

surface,

while _~c and

~cg ^are the

ordinary

and the Gaussian

bending rigidity.

^The spontaneous curvature Co is _a model parameter ^that arises if the symmetry ^{between the two}

monolayers

is broken.

Physical examples

^{for this} case are:

I)

_a ^membrane with two different

monolayers forming

a

bilayer

_or

it)

_a

bilayer facing

two different solvents. The Gaufl-Bonnet theorem states that the second _term is invariant under

transformations which do not

change

the

topology

of the surface. As

long

_as the

topology

is

fixed,

^this term _can and will be obmitted.

The

phase diagram

is determined

by minimizing

the _energy F for fixed _area A of the surface and fixed enclosed volume V. Since F is invariant under

scale-transformations,

there _are

only

two

independent parametdrs.

These _are the reduced volume

U ~

~rji13

~~~

and the reduced spontaneous curvature

co +

co1G

,

(3)

"'~~~~

jio

₊

(A/4ir)~/~

_~~~

is the radius of _a

sphere

^{with surface} _area A.

Then,

the _energy F

=

F(u,co)

and the

phase diagram depend

_{on v} and _co

only.

2.2 BILAYER-couPLE _MODEL. In the BC model (14] ^the

bending

_energy reads

G _e

f / _dA(Ci

+

C2)~ (5)

2

As _a third constraint, the _area difference

AA _w A~~

A""

_m 2DM

(6)

between the _outer and the inner

monolayer

is

kept fixed,

which _can be calculated from the total _{mean curvature}

M ₊

/ _dA(Ci

+

C2) (7)

2

and the distance D between the neutral surfaces of the

monolayers. Physically,

the constraint

on AA would describe

incompressible monolayers

which do not

exchange lipid

molecules. In this _case the energy G ₌

G(u,m)

_can be

expressed by

the reduced volume _v and the reduced total _{mean curvature}

m e

M/1~ (8)

As derived in reference

[12],

^the

stationary shapes

_are the _same in both models. The

phase

diagrams

_are,

however, completely

different.

2.3 STATIONARY _{SHAPES OF} AXIAL SYMMETRY: SHEETS OF SOLUTIONS.

Stationary shapes

of axial symmetry are determined

by solving

the

shape equations (A4-A9)

derived in

appendix

A. The solutions to these

coupled

nonlinear differential

equations

_{are a} discrete set of sheets with different

energies

F _or G.

Stationary shapes forming

such _a sheet _can be either saddle

points

_or local

minima,

I-e-

equilibrium shapes,

of the curvature energy. The distinction

between both _cases

requires

_a

complete stability analysis. However, stability

with respect to deformations

preserving axisymmetry

_can be checked

by

close

inspection

of the _energy

diagrams

as discussed in

appendix

B. In order to

study stability

with respect to

non-axisymmetric deformations,

_we use an

approximation

based _on conformal transformations _as described in the next section.

Three different sheets of

stationary shapes

_can be

distinguished,

_see

figure

1: I) the sheet of

sickle-shaped tori; it)

the sheet of discoide tori and

iii)

the sheet of toroidal stomatocytes

which do _not have _a symmetry

plane perpendicular

to the symmetry ^axis.

Both,

the discoide tori and the

sickle-shaped

tori have reflection symmetry with respect to the

plane perpendicular

to the axis of symmetry.

They

_can be

distinguished

_as ^different sheets since

they

_are

separated

from each other except ^for one

shape

which is the Clifford torus.

Therefore, starting

with the Clifford torus which has _an

exactly

circular _cross

section,

the _two different

ellipsoidal

deformations of the _contour lead to the sheets of discoide and

sickle-shaped

tori [27]. ^{The sheet of toroidal} stomatocytes ^{bifurcates from the} sheet of

sickle-shaped

tori and from the sheet of discoide tori, thus

connecting

these two sheets. As _an illustrative

example,

we show in

figure

la the _energy G of the sheets of

stationary shapes

_versus the reduced total

mean curvature m for

shapes

with fixed reduced volume _u

= 0.55. The detailed discussion of this

diagram

is

given

in

appendix

B. In

figure 16,

_we show the

corresponding

calculated

contours of

shapes

of lowest energy G.

Various types of limit

shapes,

where the sheets end and the

shape

becomes

singular,

_can be

distinguished.

For all three

sheets,

_one class of limit

shapes

_occurs, where the hole diameter vanishes.

Formally,

these

shapes

represent a connection to the

spherical topology

and _are denoted

by

Ls;~k, Ld;« and

Lsto.

Other classes of limit

shapes

include tori with

exactly

circular

cross section L~;rc ^and

shapes

with

diverging

^hole diameter. A detailed discussion of the limit

shapes

is

given

in

appendix

B.

2.4 AXIAL-SYMMETRY-BREAKING _CONFORMAL TRANSFORMATIONS. The

stationary shapes

with axial symmetry can be obtained

by solving shape equations

for the _contour. Non-

axisymmetric shapes

become accessible

through

conformal transformations

applied

to

axisym-

metric

shapes.

The usefulness of this

approximate

nlethod to determine the full

phase diagram

is first shown

using

the Clifford torus.

2.4.1 The Clil&ord _torus and its conformal transformations. The Clifford _torus is _{an ax-}

isymmetric shape

with _a circular _cross section and _a fixed _ratio of

v5

_{of its}

_generating

_radii

with reduced volume _v

= uci ^e

3/(25/~irl/~)

_ci 0.71 and the reduced total _{mean curvature}

m = mcj e

ir~/~25R

ci 13.24. The Clifford _torus is _a

stationary shape

of the _energy F for _any value of Co [28].

An

important

property ^of this

shape

_was

conjectured by

Willmore [21] ^for

genus-I surfaces,

the Clifford torus is the absolute minimum of the

bending

_energy

G _e

f

/dA(Ci

⁺

^{C2)~ (9)}

2

with G

=

Gci

_% 4ir~~c.

N°11 SHAPE TRANSFORMATIONS OF TOROmAL VESICLES 1685

2.5 Ls,ck

sweshaoed

ton ~(2)

~'~~

i§

CD _,'

~/

~sto '~ _~d'SC

dscc,d

~.° SZ©xrpes ~d,sc ^~°~

II

c*

~

l~)

~ Lanc

0.5 1.0 m/41r

~s,ck ~~~k

~d,sc ~D~

,

' '

' '

~fl~j

' '

~'~ ~

j

l~

(b)

^{0.29 0.50 0.63 0.52 1.05 1.36}

Fig-I- a)

Energy G of the axisymrnetric stationary shapes in the BC model _versus the total _mean curvature _m for fixed reduced volume _v

= 0.55. Three different branches _can be distinguished: I) _a branch of discoid tori;

it)

_a branch of sickle-shaped tori and

iii)

_a branch of toroidal stomatocytes.

The toroidal stomatocytes ^bifdrcate from the other branches at the points C))/~, C)))~ and Cd;sc.

All three branches end at different types of limit shapes: Ls;ck, Lsto and Ld;sc are limit shapes with

a vanishing hole diameter. At Lc;rc, _a limit shape with perfectly circular _cross section is reached.

Some parts ^of these axisymmetric branches _are unstable with respect to symmetry breaking ^conformal transformations. These instabilities _occur at the points C*. b) Sequence of axisymmetric shapes of minimal _energy G for fixed reduced volume _u

= 0.55 and several values of

m/4jr.

The first and the last shape _are unstable with respect to non-axisymmetric deformations.

Since the

bending

_energy G is _an invariant under conformal transformations in the three- dimensional

embedding

space, every

shape

that is _a conformal transformation of the Clifford

torus also has the _{same energy} G.

Consequently,

the _energy minimum for tori is

degenerate

_as

first observed

by Duplantier

_[22] if _no further constraints _are

imposed.

The nontrivial conformal transformations in three dimensions _are the

special

conformal transformations which _can be

parametrized by

a vector a. This transformation consists of

an inversion R _-

R/R~,

_a translation

by

_a vector _a and another inversion.

Every

point R is thus transformed to

R',

with

~

l~~~~~

_~~~~

Note,

that _two successive

special

conformal transformations

(10)

with translation vectors al

Fig.2.

The Clifford torus and

a shape generated by _a special conformal transformation _as in

(10)

with translation vector _{ax =}

0.3/Ro,

and _ay

= az = 0.

and _{a2 are}

equivalent

to _one

special

conformal transformation with _{a = ai + a2,}

I-e-,

the

special

conformal transformations form _a commutative

subgroup.

Conformal transformations of the Clifford _torus generate _a one-parameter

family

of _non-

axisymmetric shapes.

This _can be _seen

by applying

_a ^conformal transformation

(10)

to the Clifford _torus with the Z-axis _as the axis of rotational symmetry and the

X-Y-plane

_as the symmetry

plane,

_see

figure

2 _{as an}

example.

First, choose _a vector _{a =}

(lcos#,

I sin

#, 0).

Varying

its

length

I

generates

a one-parameter

family

of

non-axisymmetric shapes

with _vary-

ing

reduced volume _uci <

u(1)

< 1. This

family

of

shapes

_can be described in _a closed

analytical

form [25]. ^{For 1} =

2~Rir~/~ /(Ro(2~/~ +1)),

it ends _up at _a limit

shape

with _v

= 1. This limit

shape

consists of _a

sphere

with _an infinitesimal handle. All these

shapes

have the _same

bending

energy G~t

" 4ir~~c.

Therefore,

the handle

gives

a finite contribution

4ir~~

8ir~c to the

bending

energy, since G

= 8ir~c for _a

sphere.

On the other

hand,

if _a

special

conformal transformation with _a

=

(0,

0,

az)

is

applied

to the Clifford torus, this leads

only

to scale-transformations and

no new

shapes

_are

generated.

The latter property

depends crucially

on the circular _{cross sec-} tion of the Clifford _torus. In contrast, for _a

general

toroidal

shape

with X-Y symmetry

plane (e.g. sickle-shaped tori),

_a conformal transformation

(10)

with _a

=

(0, 0, az)

does generate a

shape

^{with broken}

up-down

symmetry.

Thus,

_even

though

the

special

^conformal transforma- tions involve the three parameters

(ax,

_ay,

az),

there _is

only

_a one-parameter

family

of

special

conformal transformations which

produces

_new

shapes

when

applied

to the Clifford torus due to its

special

symmetry

properties. Therefore,

for fixed _u, there is _no conformal

degeneracy

that would lead to conformal modes

(or

"massless Goldstone modes" [23].

2.4.2 Axial

symmetry-breaking: approximate phase

boundaries. The

non-axisymmetric

toroidal

shapes

of lowest

bending

_{energy, as} obtained via conformal transformations of the Clifford torus, represent ^the

ground-state:

I) ^for co " 0 and _{u > u~t} in the SC model and

it) along

_a line _v

=

v~(m)

in the BC model. This

indicates,

that

regions

of

non-axisymmetric shapes

exist in the

phase diagrams

of both models [29].

We _now

apply

the

hypothesis

that conformal transformations of the various sheets of axisym- metric

shapes provide

_a reasonable estimate for the variational solution for all

regions

of the

phase diagram. Specifically,

we

analyse

the infinitesimal

stability

of the

axisymmetric

sheets with respect to

special

conformal transformations which _can be calculated from the contour line of this

shape

_[30]. This method

gives

_a lower bound _on that

region

of the

phase diagram

N°11 SHAPE TRANSFORMATIONS OF TOROIDAL VESICLES 1687

Fig.3.

An axisymrnetric sickle+haped torus with _v

= 0.55 and

m/4jr

= o-S and _a

non-

axisymmetric sickle-shaped torus. The non-axisyrnmetric shape _was generated by applying the special conformal transformation

(lo)

with _aX

"

0.3/Ro,

and _ay

= az " 0 to the axisymrnetric shape.

in which the

ground

state is

non-axisymmetric.

The idea behind this method is to start with

a

shape Si

_on the sheet of lowest _energy

shapes

and

apply

_a

special

^conformal transformation to Si with _aX

#

0, _see

figure

3 for _an

example.

This transformation generates _a new non-

axisymmetric shape, Sl',

with the _same

bending

_{energy as} Si which is located _{on a} different

point

in the

phase diagram.

The

stability

of the

original shape

_now

depends

_on the difference of the _energy of

Sl'

and the _energy of the

axisymmetric shape

S~~~ which is located _at the

same

point

in the

phase diagram.

^The

complete

derivation of this

stability

criterion is

given

in

appendix

C.

2.5 PHASE DIAGRAM oF THE SPONTANEOUS-CURVATURE MODEL. The

analysis

of the

stationary axisymmetric shapes together

with the

stability analysis

with respect to conformal transformations leads to

phase diagrams. Quite generally,

a

phase diagram

consists of several

regions

where the

shapes

have different symmetry

properties.

Transformations between

shapes

of different

regions

_are called

shape

transitions.

The

phase diagram

for the SC model is shown in

figure

4. It contains three

regions: I)

a

region

of

axisymmetric sickle-shaped

tori;

it)

_a

region

of tori with

nearly

circular _cross section

"circular tori" [31] ^and

iii)

_a

large region

of

non-axisymmetric shapes.

The _two

axisymmetric regions

are

separated by

the discontinuous

phase boundary Dax.

The

axisymmetry

of the sickle

shaped

tori and the circular tori is broken

continuously

at the lines C];~~ and C(~~~,

respectively.

These lines end _{up at two} different critical

endpoints Di

and

D2

_on the discontinuous transition line Dax. The line Dax extends into the

non-axisymmetric region

^{until it} ends _up in the critical

point D~p.

^{The Clifford} torus is located

along

a line CL with _v

= u~i which connects the lines C(;~~ ^and Dax.

The lines C]~~~ ^and C(;~~

give

the instabilities with respect to

axisymmetry breaking

^if

deformations _are restricted _to conformal transformations.

Thus, they

represent bounds for the instabilities which would have been obtained

by

_a full

stability analysis

of the

axisymmetric

sheets: the

non-axisymmetric regions

extend at least to the lines C]~~~ ^and C]~~~. ^For the Clifford torus, ^the conformal mode is the unstable

eigenmode.

To get an idea of the

shapes

in the

non-axisymmetric region

close to C(;~~, a

conformally

unstable axisymmetric stationary

shape

can be used. By

applying

_a conformal transformation to such _a

shape, non-axisymmetric

shapes

_are

generated

which should be

good approximations

for

shapes

of minimal _energy in

2

'

~/ ',

Co

/ _~~~ IQ

'

~j

~~~~~~

~'

'

~

'

i~

'

"

~

' ,

nonaxisymmetnc sickle-shaped for,

-2

0 0.5 _v

Fig.4.

Phase diagram for toroidal vesicles in the SC model. The two model parameters are the reduced volume _u and the reduced spontaneous curvature co- The line Dax which ends in the critical

point Dcp, represents the line of discontinuous shape transformations. It separates _a region of circular tori from _a region of sickle-shaped tori. At the dotted line SIS;ck, the

sickle-shaped

tori selfintersect.

The two dashed lines C];~k ^and C(;~~ represent ^lines ofinstabiliy of axisymmetric shapes with respect to symmetry breaking conformal transformations. They _{serve as} approximations for the exact continuous

shape transformations of axisymmetry breaking. Two different types of non-axisymmetric shapes _can be distinguished: _I) non-axisymmetric sickle-shaped tori and it) non-axisymmetric circular tori. For

co " 0, the Clifford _torus

(CL)

_appears in the phase diagram _on the line C],~~. ^{The Clifford} torus is

the minimal _energy shape along the dotted line with

v = u~j.

this

region.

An

example

for _a

shape generated

ⁱⁿ this way is the

conformally

transformed Clifford _torus shown in

figure

2.

Likewise,

_a

conformally

unstable axisymmetric

sickle-shaped

torus can be used to

approximate non-axisymmetric sickle-shaped

tori below C]~~~, see

figure

3.

The lines

Mi

and M2 represent borders of

metastability

related to the discontinuous transi- tions

along

the line Dax.

Starting,

for

example,

from the

region

of

sickle-shaped

tori,

crossing

the line

Dax,

the

sickle-shaped

tori become metastable while the circular tori _are

shapes

of min- imal energy. At

Mi,

the metastable

sickle-shaped

tori become unstable.

Similarly,

metastable

circular tori become unstable at M2.

Along

the line SIsj~k the contour of the

sickle-shaped

tori selfintersects in the symmetry

plane

of these

shapes.

For _a calculation of

shapes beyond

this line, one would have to include selfinteractions of the vesicle membrane.

2.6 PHASE _DIAGRAM oF THE BILAYER-couPLE MODEL.

Figure

5 shows the

phase diagram

of the BC model which

depends

_on ^the reduced volume _v and the reduced total _{mean curvature}

m. For _a closed

surface,

the surface average of the _mean curvature is

positive

and therefore

m > 0. Four different

regions

_can be

distinguished:

I) a

region

^of discoid

tori; it)

_a

region

of

sickle-shaped tori; iii)

_a

region

^{of toroidal} stomatocytes ^and

iv)

_a

large region

^of non-

axisymmetric tori.

N°11 SHAPE TRANSFORMATIONS OF TOROIDAL VESICLES 1689

2

,

~ l

~

@

j ,

'

~

0

ig. 5. -

eparatedby shape

transformation fines Csick and

Cd;sc can be distinguished: I) discoid

tori; it) sickle-shaped tori and iii) _toroidal tomatocytes. The _line C* indicates

the instability respect

to

axisymmetry _breaking

conformal On the ^right hand side of this line, non-

axisymmetric shapes have nfinimal energy. The lines_Ldisc> Lsto and Ls,ck

represent

limit _{shapes with} vanishing hole diameter. Above _the line of _circular limit shapes Lcirc, no axisymmetric stationary

shapes exist. The contour of discoid tori begins to selfintersect at _the line SIdisc.

The mirror symmetry

planes

of the discoid tori and of the

sickle-shaped

tori _are broken _con-

tinuously

at the lines Cdi« and Csj~k,

respectively, leading

to the

region

of toroidal stomatocytes without _a reflection symmetry

plane.

Continuous

axisymmetry breaking

occurs

along

the line C* which separates a

region

of _non-

axisymmetric shapes

for

relatively large

_v in the

phase diagram

from the

region

of

axisymmetric shapes

for

relatively

small _v [32]. ^{This line}

begins

at

large

m with discoid tori with _a

nearly

circular _cross section. It continues with

sickle-shaped

tori until it ends in the

point iv, m)

₌

(0,0).

A

special shape

_on this line is the Clifford torus,

again

denoted

by

CL. In this

phase diagram,

the Clifford torus is _a multicritical point ^{where the} continuous

shape

transformation lines

Cdi«,

C~i~k and C* meet.

The line C* _was obtained

by analyzing

the conformal

instability

of

axisymmetric shapes.

This conformal

instability

which _occurs in the BC model for the _same

shapes

_as in the SC

model,

is

again

_a lower bound for the extension of the region of

non-axisymmetric shapes.

It becomes the

correct

phase boundary

at the Clifford torus CL. The

examples

of non-axisymmetric

shapes

as shown in

figures

2 and 3 hold for the BC model _too.

The dotted lines in the

phase diagram

represent lines of limit

shapes

and

selfintersection,

_as described in

appendix

B. A

general

property of the

phase diagram

of the BC model

is,

that all

shape

transformations _are continuous _as it has

previously

been observed for

spherical shapes

[12]. However, we are not _aware of any reason for the absence of first order transitions in the

BC model for all

topologies.

3. Relation to

experiments:

The ADE-model.

3.1 THE ADE-MODEL. The two curvature models described _so far _{seem to contain} essential features of the

bilayer

membrane. Reflection _on the

physical

assumptions

underlying

these

models, however,

reveals that both of them

give only

_a ^crude

representation

of _one

important

aspect ^{of the}

physical properties

of the

bilayer

system. The two

monolayers forming

_a fluid

lipid bilayer

_are able _to slide with respect to each other with relative _ease. When such _a

bilayer

forms _a closed vesicle and

experiences changes

in local curvatures, ^there will be _{a net}

expansion

_or

compression

of the surface _area of the individual

monolayers

of the order

D/Ro.

Since relative movement can take

place

between the

leaves,

such

expansion

_or

compression

will be

homogeneous

_over the whole surface _or "non-local" rather than location

dependent

[11]. ^{Neither the SC model} nor the BC model has

incorporated

this effect. The SC model

neglects

this non-local

factor,

since it

implicitly

assumes that the _two

monolayers

_are

rigidly

connected and have _no relative movement, ^thus

being essentially

_a model for

interdigitated monolayers.

On the other

hand,

the BC model

recognizes

the

specific

_way of

coupling

between the two

monolayers

but does _not allow _any deviation of the

monolayer

surface _area from its

equilibrium value, assuming

_a zero

area-compressibility

for the individual

monolayers.

We will _now consider _a model where this non-local

bending

_energy is

incorporated.

In

addition,

it will be assumed that the number of

lipid

molecules remains constant within each

monolayer,

I-e-, that _no

exchange

^of

lipid

molecules _occurs between the two

adjacent monolayers

even

though

their

density

is _now different. This is the

area-difference-elasticity (ADE)

model.

Its _energy reads in scaled variables

[15,

16, 18]

W _e ^~

/(Ci

⁺

^C2)~dA

⁺

^aim _mo)~l"

^{G + ~"}

^{(m mo)~ Ill)}

2 2

The second term

explicitly

attributes _an elastic _energy to deviations of the _area difference AA _e 2DRom from _an

equilibrium

value AAO + 2DRomo. The _area difference AAO of the relaxed

bilayer depends

_on the

preparation

of each individual vesicle. The dimensionless

parameter a is the ratio of _area difference elastic

energies

to the usual

bending energies

and is for realistic

bilayers

close to _one. For the calculation of the

phase diagram,

_we choose _a

= 1

which is

supported by

the _so far

only

measurement of this

quantity [18].

^{In the limits of}

large

a and small _a, the BC model and the SC model _are

recovered, respectively.

The phase

diagram

of the ADE-model _can be derived from that of the BC model _as follows:

the energy W has to be minimized for fixed enclosed volume V, surface _area A and _mo.

Stationary shapes

of the _energy W of the ADE-model _can be determined

by

first

varying

W with respect to the

subspace

of

shapes

with fixed reduced total _mean curvature _m. This leads to the _energy

W(u,

_m,

mo)

"

G(u, m)

+

~ca(m mo)~/2 (12)

The final variation of

W(v,

_m,

mo)

with respect to _m for fixed _mo leads to

~~jj>

^~~ =

-J~aim mo) i13)

This relation determines the value of _{m as a} function of _mo.

Inserting

this value of _m in

equation (12)

leads to the energy

W(mo>u)

of

stationary shapes

of the ADE-model. The

curves of conformal

instability

in the BC model _can be transformed

by

the _same

generalized

Legendre

transformation to the ADE-model since the conformal

stability

of

axisymmetric shapes

with mirror symmetry

plane

does _not

depend

_{on a} [33].

In

figure 6,

_we show the

phase diagram

of toroidal vesicles in the ADE-model. It

depends

on two parameters, ^{the reduced volume} u and _mo. This

phase diagram

is

qualitatively

sim-

N°11 SHAPE TRANSFORMATIONS OF TOROmAL VESICLES 1691

2

0

0.5 v

Fig.6.

Phase

diagram

for toroidal vesicles in the ADE-model with _{a =} 1. The phase diagram

is very similar to that of the BC model elplained in figure 5. Three axisymmetric regions which _are separated by continuous shape transformation lines Csick and Cdisc _can be distinguished: _I) discoid tori;

it)

sickle-shaped tori and iii) toroidal stomatocytes. The line Ldisc represent limit shapes with vanishing

hole diameter. The instability with respect to axisymmetry

breaking

conformal transformations is

denoted by C*. Within the region of axisymmetric shapes with reflection plane, discontinuous shape transformations from circular tori to discoid tori _occur along the line D. This line ends up in _a critical

point Dcp. ^{The Clifford} torus is the shape of minimal _energy along the dotted line CL. A temperature change corresponds to _a temperature trajectory _as indicated by the dashed line

(tt).

This trajectory

crosses the line C* which implies that axisymmetry breaking shape transformations of toroidal vesicles

can be induced by temperature changes.

ilar _to the

phase diagram

^{of the BC}

^model,

compare

figure

5. There is _a

large region

of

non-axisymmetric shapes

which _are

separated

from the

axisymmetric shapes by

the

phase boundary

^{C*. All sheets of}

axisymmetric shapes

_appear within the ADE-model _as

shapes

of

minimal _energy: I) ^discoid

tori; it) sickle-shaped

tori and

iii)

^toroidal stomatocytes.

Symmetry breaking shape

transformations _occur at the lines

C*,

Cdi« and Csi~k. These transitions _are all continuous. A discontinuous transition line D which ends in the critical point D~p separates discoid tori from tori with _a circular _cross section within the

region

of mirror

symmetric

_ax-

isymmetric shapes.

While in the BC model the Clifford torus is _a multicritical

point

where all symmetry

breaking phase

boundaries meet, a whole line CL of Clifford tori exists in the

phase diagram

^{of the}

ADE-model,

_as it does in the SC model.

3.2 EXPERIMENTALLY OBSERVED SHAPES: LOCATION IN THE PHASE DIAGRAM. The obser-

vations

by Fourcade,

Mutz and Bensimon include

shapes

of _{genus one} and _{genus two.} Vesicles of _{genus one can} be

immediately

located in the

phase diagram

for toroidal vesicles in the

ADE-model. As _a first

example,

consider the

axisymmetric

circular _torus shown in reference [8] for which the authors estimate _{u ci} 0.5. The value of _mo cannot be measured

directly.

In

principle,

it could be obtained

by carefully comparing

the contour of the observed torus with

the solutions of the

shape equations. Inspection

of the

phase diagram shows,

that circular

shapes

^with _{u =} 0.5 exist within _a

large

_range of _mo, indeed.

Second,

consider the observed

non-axisymmetric

torus with _{u ci} 0.77. This

shape

resembles the

non-axisymmetric shape

shown in

figure

2. The

phase diagram

of the ADE-model shows that

shapes

with _{u >} 0.77 _are

definitely non-axisymmetric

for all _mo. ^'~

Let _{us now} discuss the

genus-2 shapes

_seen

by

Fourcade _et al.. Even without _an

explicit

calculation of minimal energy

shapes

with _{g >} I, ^which are

necessarily non-axisymmetric,

the observed

genus-2 shapes

_can be related _to calculated

genus-I shapes

which _occur in the

phase diagram.

This _can be _seen

by starting

_e-g- ^with a discoid torus in the limit where the hole diameter vanishes

ii-e-

^with a limit

shape

Ldi« _as discussed in

appendix B).

In this

case, the hole

gives

_no contribution to the

bending

_energy and the

shape

is identical to _a

spherical discocyte.

^If _{now a} second infinitesimal hole is

inserted,

the energy will

only change infinitesimally.

This indicates that

mirror-symmetric

discoid

shapes,

with two holes in this symmetry

plane,

_occur in the

phase diagram

of

genus-2

^vesicles [34].

They

_are related to discoid tori of _{genus one.} An

example

of such _a discoid

shape

of _genus 2 has been

reported

ⁱⁿ

reference [8].

By

_a similar argument

sickle-shaped

tori _can be related to the second type of genus-2

shapes

observed _so far. In the limit of _{zero u,} the

sickle-shaped

tori consist of _two concentric

spheres

of

equal

radii connected

by

two infinitesimal necks.

Inserting

a third infinitesimal neck to _a

sickle-shaped

torus with infinitesimal _u does not

change

the energy. This indicates that

shapes consisting

^of two

spherical

parts connected

by

three necks _are

genus-2 shapes

with minimal

energy for small _u and small _m. A third neck breaks the

axisymmetry

of the

original

sickle-

shaped

tori, but _can be

arranged

in such _a way that the positions of the necks

obey

_a threefold symmetry. Conformal transformations

applied

to such _a

shape

will induce _a shift of the neck

positions comparable

to the _one shown in

figure

3 for the

genus-I

torus. In

fact,

_a

shape

with

fluctuating

neck

positions

has

recently

been observed [9].

3.3 PREDICTIONS FOR FUTURE EXPERIMENTS. So

far,

in the experiments

by

Bensimon

and co-workers

shapes

with non-trivial

topology

_are recorded without

changing

external _pa- rameters. For _a crucial test of the

theory presented here,

_one should _{vary one} parameter sys-

tematically. Temperature changes

have

successfully

been used for vesicles of

spherical topology

to induce

shape

transitions [4, 5]. ^{We recall} that _a

change

in temperature ^{affects both} u and mo because of the thermal

expansivities

of the

monolayers

of the membrane and the enclosed

water. These temperature

trajectories

in the

phase diagram,

and the

corresponding

_sequence

of

shape transformations,

_are

significantly

affected

by

_any small asymmetry ^of the order of 10~~ _{in the} thermal

expansion

coefficients of the two

monolayers

_[4,

12].

^So

far,

there _are

no

independent

measurements _or estimates for such _a

possible

_asymmetry. In this _{paper, we} therefore discuss the ideal _case of

symmetric

thermal

expansion

of the

monolayers.

If the total

bilayer

volume is constant, temperature

trajectories

_can be described

by

_[12]

mo "

(mo@)/u

,

(14)

where

if, mo)

denotes _a

starting point

in the

phase diagram. Figure

6 shows such _a tem-

perature

trajectory (tt).

For

decreasing

temperature the reduced volume _u increases while mo decreases.

Starting

in the

axisymmetric region

^{of discoid}

tori,

^the

trajectory

_crosses the continuous

shape

transformation C* where the axisymmetry ^{is broken.} Thus, the model pre- dicts that for any circular torus a decrease in temperature will lead to a

non-axisymmetric

torus via _a continuous

shape

transformation

(provided

the

bilayer

remains in the fluid

state).

Since this transformation is reversible,

increasing

^the temperature, I-e-

reducing

_u, renders